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    <article id="post-图" class="article article-type-post" itemscope itemprop="blogPost">
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    <a href="/2019/08/04/图/" class="article-date">
  <time datetime="2019-08-04T03:47:01.000Z" itemprop="datePublished">2019-08-04</time>
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      <a class="article-title" href="/2019/08/04/图/">图</a>
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        <h1 id="1-图的表示方法"><a href="#1-图的表示方法" class="headerlink" title="1. 图的表示方法"></a>1. 图的表示方法</h1><pre><code>图的数学表达式为 G=(V,E), 解释为图 G 中是一组由 |V| 个列表的数组　Adj 组成 , 其中的每一个列表对应着这个 V 中的一个节点 . 对于每一个 u ∈ V , 邻接表 Adj[u] 包含所有的满足条件 (u,v) 
∈ E 的顶点 u (Adj[u] 中包含图 G 中所有和顶点 u 相邻的顶点). 

图可以分为有向图和无向图 , 主要的区别在于连接顶点的边是否有方向的矢量特性:
</code></pre><p><img src="/image/Selection_001.png" alt="avatar" title="有向图和无向图"> </p>
<pre><code>表示图的方式方法有两种 , 分别为邻接表和邻接矩阵:
a. 邻接表是由包含 |V| 个列表的数组组成，其中每个列表对应这图中的每个顶点，列表中存储着图中和当前顶点相邻的顶点 : 
</code></pre><p><img src="/image/Selection_002.png" alt="avatar" title="邻接表"> </p>
<pre><code>b. 邻接矩阵是存储这顶点间的相邻关系, 如果 A[i,j] == 0, 表示顶点 i 和顶点 j 不相邻 , 如果 A[i,j] == 1, 表示顶点 i 和顶点 j 相邻
</code></pre><p><img src="/image/Selection_003.png" alt="avatar" title="邻接矩阵"> </p>
<pre><code>关于图的基本概念:
a. 无向图中，所有的邻接表的长度之和为 2|E| , 有向图中，　所有邻接表的长度之和为 |E| .
b. |E| &lt;&lt; |V|^2 ,　为稀疏图 . |E| ~= |V|^2 ,  表示稠密图 . 
c. 加权图: 顶点和顶点间的边不在为１，而是根据需求表示加权值.
</code></pre><h2 id="广度优先搜索"><a href="#广度优先搜索" class="headerlink" title="广度优先搜索"></a>广度优先搜索</h2><pre><code>    广度优先搜索会始终将已发现和未发现之间的边界，沿着广度的方向向外扩展 . 意味着首先会发现距离给定原点 s 的距离为 k 的所有顶点 , 然后才会发现离 s 距离为 k+1 的其他顶点 . 在广度优先
搜索的过程中 ， 距离给定原点 s 为 k 是距离 s 距离为 k+1 顶点的长辈节点， 祖辈关系组成广度优先树.

<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br></pre></td><td class="code"><pre><span class="line">广度优先算法 - BFS                 # 算法中 π[u] 表示顶点 u 的父亲节点, color[u] 表示顶点 u 的颜色. d[u] 表示距离顶点 s 的距离</span><br><span class="line">BFS (G, s)  </span><br><span class="line">    for u in V[G]   </span><br><span class="line">        color[u] ← Withe    </span><br><span class="line">        d[u]     ← ∞        </span><br><span class="line">        π[u]     ← NIL           # 将图中所有顶点的 p[u] , color[u] , d[u] </span><br><span class="line"></span><br><span class="line">color[s] ← GRAY </span><br><span class="line">d[s] ← 0    </span><br><span class="line">π[s] ← NIL                       # 初始化顶点 s</span><br><span class="line"></span><br><span class="line">Q ← NIL </span><br><span class="line">ENQUEUE (Q, s)                   # 将顶点 s 插入队列 Q</span><br><span class="line"></span><br><span class="line">while Q != NIL                   # 结束条件为队列 Q 为空时，跳出循环</span><br><span class="line">    do u ← DEQUEUE (Q)           # 将队列 Q 出列一个顶点 u, 然后把顶点 u </span><br><span class="line">    for each v ∈ Adj[u]          # 的邻接表遍历，如果邻接数组中的顶点的颜色</span><br><span class="line">    </span><br><span class="line">        do if color[v] == White  # Withe , 设置顶点 u 为 u 邻接数组中顶点 (v)</span><br><span class="line">            then color[v] ← GRAY # 的父亲节点，则顶点 v 设置为 GRAY。并将顶点 v </span><br><span class="line">            d[v] ← d[u] + 1      # 入队列 Q</span><br><span class="line">            π[v] ← u</span><br><span class="line">            ENQUEUE (Q, v)</span><br><span class="line">    color[u] ← BLACK             # 设置顶点 u 为 BLACK</span><br></pre></td></tr></table></figure>
</code></pre><h3 id="最短路劲"><a href="#最短路劲" class="headerlink" title="最短路劲"></a>最短路劲</h3><pre><code>  对于图 G=(V,E), 广度优先搜索算法可以得到从已知源顶点 s∈V 到每个可达顶点的距离. 顶点 s 到顶点 v 的最短距离 δ(s,v) 是从 s 到 v 的任何路劲中边数最少的路径
如果 s 到 v 没有通路， 则 δ(s,v) = ∞ ;

定理 1 : 设 G = (v, E) 是一个有向图或者无向图， s∈V 为 G 的任意一个顶点，对任意边 (u, v) ∈ V ,有
        δ(s,v) ≤ δ(s, u) + 1
定理 2 : 图 G = (V, E) 是一个无向图或有向图，假设算法 BFS 从 G 中某一个给定源顶点 s∈V开始执行。在终止执行时，　则有 
        d[v] ≥ δ(s,v)
定理 3 : 在图 G = (V ,E) 上执行 BFS 的过程中，队列 Q 中包含顶点 &lt;v₁, v₂ ..., vr&gt; , v₁为队列 Q 的头，vr 为队列 Q 的尾，则有
        d[vr] ≤ d[v1] + 1 , 且 d[vi] ≤ d[vi+1]
定理 4 : 在执行 BFS 过程中将顶点 vi 和 vj 插入了队列， 且 vi 先于 vj 入队，那么当 vj 入队时有, 
        d[vi] ≤ d[vj]
定理 5 : 在执行 BFS 从给定 G 中某一个顶点 s∈V 开始运行。那么在执行的过程中，BFS 可以发现源顶点 s 可以到达的每个顶点 v∈V开始运行。在
        运行终止时，对所有 v∈V , 都有 d[v] = δ(s,v)

广度优先搜索树是 BFS 在图 G 上运行后的 p 域组成。对于图 G = (V, E), 给定的源点 s, 可以更为形式地定义其前趋子图: Gπ = (Vπ, Eπ), 其中:
    Vπ = {u∈V: π[u] != NIL} U {s}
且
    Eπ = {(π[v], v): v∈ Vπ - {s}}
</code></pre><h2 id="深度优先搜索-DFS"><a href="#深度优先搜索-DFS" class="headerlink" title="深度优先搜索 (DFS)"></a>深度优先搜索 (DFS)</h2><pre><code>1. 深度优先搜索的过程: 给定图 G = (V,E), 从任一顶点 v 开始扫描 v 邻接表 adj[v], 顶点 u 属于 adj[v] 。则下一次扫描从顶点 u 开始 , 重复顶点 v 的搜索过程. 
                       如果还存在为发现的顶点, 以这个顶点作为源点做 DFS.

2. 深度优先森林: 在做 DFS 的过程中 , 扫描顶点 v 的邻接表 adj[v] , 顶点 u ∈ adj[v] 且是第一次被搜索. 则顶点 u 的先辈子域 π[u] = v . 由先辈子域 π[] 组成的
                 树称为深度优先树。在 DFS 过程中会生成多颗深度优先树，多颗深度优先树组成深度优先森林。
                G = (V, Eπ) , Eπ = ((π[v], v), v∈ V &amp; Vπ ~= NIL)

3. 时间戳: 顶点 v 在第一次被发现第一次记录下第一个时间戳 d[v] , 在结束搜索时记录下第二个时间戳 f[u] . 用来推测 DFS 的进行情况 .

4. DFS 的伪代码

<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br></pre></td><td class="code"><pre><span class="line">DFS (G)</span><br><span class="line">for each vertex u ∈  V[G]   # 此处是初始化所有顶点的 color 和 π 域</span><br><span class="line">  color[u] ←  WHITE</span><br><span class="line">  π[u]     ←  NIL</span><br><span class="line"></span><br><span class="line">time ← 0                    # time 用来记录时间戳, 为全局变量</span><br><span class="line"></span><br><span class="line">for each vertex u ∈  V[G]   # 此处为每个顶点做 DFS </span><br><span class="line">  do if color[u] == WHITE</span><br><span class="line">    then DFS-VISIT(u)</span><br><span class="line"></span><br><span class="line">################################################################################</span><br><span class="line">DFS-VISIT (u)</span><br><span class="line">color[u] ← GRAY              # 顶点 u 第一次被搜索，将顶点 u 的 color 域置为灰色 (GRAY)</span><br><span class="line"></span><br><span class="line">time ← time + 1              # 记录下顶点 u 的开始的时间戳</span><br><span class="line">d[u] = time </span><br><span class="line"></span><br><span class="line">for each v ∈  adj[u]         # 遍历顶点 u 的邻接表 adj[u] ，如果顶点 v 的 color 域为</span><br><span class="line">  do if color[v] == WHITE    # WHITE , 则设置顶点 v 的先辈域 π[v] = u</span><br><span class="line">    then π[v] ← u</span><br><span class="line">      DFS-VISIT (v)</span><br><span class="line"></span><br><span class="line">color[u] ←  BLACK</span><br><span class="line">f[u] ← time ← time + 1      # 记录下顶点 u 的结束的时间戳</span><br></pre></td></tr></table></figure>

5. 图中边的分类
  a. 树边: 顶点 v 在探索 (v,u) 是首次被发现的，那么 (v,u) 是一条树边
  b. 反向边: 连接到顶点 u 到他某一祖先顶点 v 的那些边。
  c. 正向边: 连接顶点 u 到某个后裔的顶点 v 的边 .
  d. 交叉边: 一个顶点不是另外一个顶点的祖先
</code></pre><hr>
<p>δ:d<em><br>∈:(-<br>∋:)-<br>π:p</em></p>

      
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    <article id="post-IPv6 Frame Format" class="article article-type-post" itemscope itemprop="blogPost">
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        <p>b - bits<br>B - bytes</p>
<h1 id="Ether-II-数据格式"><a href="#Ether-II-数据格式" class="headerlink" title="Ether II 数据格式"></a>Ether II 数据格式</h1><pre><code>| Dest Mac   | Sou Mac    |  type  |
+------------+------------+--------+----------+
| 12 b       |    12 b    |  8 b   | payload  |
+------------+------------+--------+----------+
</code></pre><h1 id="IPv6-头格式"><a href="#IPv6-头格式" class="headerlink" title="IPv6 头格式"></a>IPv6 头格式</h1><pre><code>+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|Version|    Traffic Class      |       Flow Label              |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|     Payload Length            | Next Header  | Hop Limit      |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
+                                                               +
|                                                               |
+                   Source Address                              +
|                                                               |
+                                                               +
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
|                                                               |
+                                                               +
|                                                               |
+                   Destination Address                         +
|                                                               |
+                                                               +
|                                                               |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
</code></pre><h1 id="IPv6-扩展头格式"><a href="#IPv6-扩展头格式" class="headerlink" title="IPv6 扩展头格式"></a>IPv6 扩展头格式</h1><pre><code>+---------------+------------------------
| IPv6 header   | TCP header + data     
|               |
| Next Header = |
|      TCP      |
+---------------+------------------------


+---------------+----------------+------------------------
| IPv6 header   | Routing header | TCP header + data 
|               |                |
| Next Header = | Next Header =  |
|   Routing     |    TCP         |
+---------------+----------------+------------------------

+---------------+----------------+-----------------+-----------------
| IPv6 header   | Routing header | Fragment header | fragment of TCP
|               |                |                 | header + data 
| Next Header = | Next Header =  | Next Header =   |
|  Routing      |     Fragment   |  TCP            |
+---------------+----------------+-----------------+-----------------
</code></pre>
      
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        <h1 id="IPv6-的概述"><a href="#IPv6-的概述" class="headerlink" title="IPv6 的概述"></a>IPv6 的概述</h1><pre><code>ipv6 采用 128 bit 的地址长度　。来代替现在地址资源即将枯竭的　ipv4 。
</code></pre><h1 id="2-IPv6-地址"><a href="#2-IPv6-地址" class="headerlink" title="2 IPv6 地址"></a>2 IPv6 地址</h1><pre><code>ipv6 采用　128 bit 的地址长度 , 格式为 
</code></pre>
        
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    <article id="post-以太网帧-格式" class="article article-type-post" itemscope itemprop="blogPost">
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        <h1 id="以太网帧的数据格式"><a href="#以太网帧的数据格式" class="headerlink" title="以太网帧的数据格式"></a>以太网帧的数据格式</h1><pre><code>以太网网帧有四种类型 ，分别为 Ethernet I, Ethernet II, NoveIIEther, EtherSNAP 。  
</code></pre>
        
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<li><p>加密 pdf 去除密码</p>
<p> $ sudo apt-get install qpdf<br> $ qpdf –password=’123456’ –decrypt in.pdf out.pdf</p>
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        <h1 id="散列表"><a href="#散列表" class="headerlink" title="散列表"></a>散列表</h1><pre><code>1. 散列表是普通数组概念的推广，可以对数组直接进行寻址，可以在 Ѳ(1)时间内访问数组的内任意的元素。
2. 散列表采用的数组的尺寸和要存储的关键字的数量成正比。
3. 散列的下标是关键字经过计算得出的。
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